Definition 3.1 .

If x 𝑥 x italic_x , y 𝑦 y italic_y , and z 𝑧 z italic_z are distinct points in a Busemann G 𝐺 G italic_G -space and

d ⁢ ( x , y ) + d ⁢ ( y , z ) = d ⁢ ( x , z ) 𝑑 𝑥 𝑦 𝑑 𝑦 𝑧 𝑑 𝑥 𝑧 d(x,y)+d(y,z)=d(x,z) italic_d ( italic_x , italic_y ) + italic_d ( italic_y , italic_z ) = italic_d ( italic_x , italic_z )

we say that y 𝑦 y italic_y lies between x 𝑥 x italic_x and z 𝑧 z italic_z and denote this by x - y - z 𝑥 𝑦 𝑧 x-y-z italic_x - italic_y - italic_z .

Definition 2 .

Let h ℎ h italic_h be a formal powerseries of the form h ⁢ ( x ) = x + ∑ k = m ∞ h k ⁢ x k ℎ 𝑥 𝑥 superscript subscript 𝑘 𝑚 subscript ℎ 𝑘 superscript 𝑥 𝑘 h(x)=x+\sum_{k=m}^{\infty}h_{k}x^{k} italic_h ( italic_x ) = italic_x + ∑ start_POSTSUBSCRIPT italic_k = italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , h m ≠ 0 subscript ℎ 𝑚 0 h_{m}\neq 0 italic_h start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ≠ 0 . Its iterative logarithm is the unique formal powerseries j 𝑗 j italic_j of form j ⁢ ( x ) = ∑ k = m ∞ j k ⁢ x k 𝑗 𝑥 superscript subscript 𝑘 𝑚 subscript 𝑗 𝑘 superscript 𝑥 𝑘 j(x)=\sum_{k=m}^{\infty}j_{k}x^{k} italic_j ( italic_x ) = ∑ start_POSTSUBSCRIPT italic_k = italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT with j m = h m subscript 𝑗 𝑚 subscript ℎ 𝑚 j_{m}=h_{m} italic_j start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT that satisfies the Julia equation

(17) j ∘ h = h ′ ⋅ j . 𝑗 ℎ ⋅ superscript ℎ ′ 𝑗 \displaystyle j\circ h=h^{\prime}\cdot j. italic_j ∘ italic_h = italic_h start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⋅ italic_j .
Definition 5.2 .

The Jordan triple product in an associative algebra A 𝐴 A italic_A over a field of characteristic not 2 2 2 2 is the trilinear operation

( a , b , c ) = a ⁢ b ⁢ c + c ⁢ b ⁢ a . 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑐 𝑏 𝑎 (a,b,c)=abc+cba. ( italic_a , italic_b , italic_c ) = italic_a italic_b italic_c + italic_c italic_b italic_a .
Definition 5 .

A complex n 𝑛 n italic_n -dimensional manifold satisfies the ( n - 1 , n ) 𝑛 1 𝑛 (n-1,n) ( italic_n - 1 , italic_n ) -th weak ∂ ⁡ ∂ ¯ ¯ \partial\bar{\partial} ∂ ¯ start_ARG ∂ end_ARG -lemma if for its every real ( n - 1 , n - 1 ) 𝑛 1 𝑛 1 (n-1,n-1) ( italic_n - 1 , italic_n - 1 ) -form φ 𝜑 \varphi italic_φ such that ∂ ¯ ⁢ φ normal-¯ 𝜑 \bar{\partial}\varphi ¯ start_ARG ∂ end_ARG italic_φ is a ∂ \partial ∂ -exact form, there exists an ( n - 2 , n - 1 ) 𝑛 2 𝑛 1 (n-2,n-1) ( italic_n - 2 , italic_n - 1 ) -form ψ 𝜓 \psi italic_ψ such that

(1.1) ∂ ¯ ⁢ φ = i ⁢ ∂ ⁡ ∂ ¯ ⁢ ψ . ¯ 𝜑 𝑖 ¯ 𝜓 \bar{\partial}\varphi=i\partial\bar{\partial}\psi. ¯ start_ARG ∂ end_ARG italic_φ = italic_i ∂ ¯ start_ARG ∂ end_ARG italic_ψ .